General Existence Principle for Singular BVPs and Its Application

Abstract:
We present a general existence principle which can be used for a large class of
singular boundary value problems of the form $$u^{(n)}(t)=f\big(t,u(t),\dots,u^{(n-1)}(t)\big),
\quad u\in \s,$$
where $f$ satisfies the local Carath\'{e}odory conditions on $[0,T] \times \D$,
a set $\D\subset \R^n$ is not closed, $f$ has singularities in its phase
variables on the boundary $\partial \D$ of $\D$, and $\s$ is a closed subset in
$C^{n-1}([0,T])$. The proof is based on the regularization and sequential
techniques. An application of the general existence principle to singular
conjugate $(p,n-p)$ BVPs is also given.